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**Unformatted text preview: **4. A point P is picked at random (i.e. uniformly with respect to area) on the surface of the unit sphere x 2 + y 2 + z 2 = 1 of Euclidean three space. Then a point Q is picked at random (uniformly with respect to length) on the line which connects the origin and P. (a) Let ( X,Y,Z ) be the rectangular coordinates of Q . Find the joint density of ( X,Y,Z ). (b) Show that the random variables X,Y,Z of part (a) are not independent. Suppose P is picked as above, and that a point R is picked so that it is on the half line starting at the origin and going through P on out to innity, with the distance of R from the origin being a random variable with a density function f . (So, the special case where f equals one between 0 and 1 gives the point in part (a).) Let ( X ,Y ,Z ) be the rectangular coordinates of R . Is there any choice of f which makes X ,Y ,Z independent? 1...

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